论文标题
椭圆操作员和应用的广义Carleson扰动
Generalized Carleson perturbations of elliptic operators and applications
论文作者
论文摘要
我们向两个方向扩展了与椭圆形真实二阶发散形式相关的dirichlet问题扰动的概念（可能是退化，不一定是对称的）椭圆操作员。首先，除了我们称之为加法的Carleson扰动的Carleson类型的经典扰动外，我们还引入了标量 - 刺激性和反对称的Carleson扰动，这两种扰动都允许在边界上进行非平凡的差异。其次，我们考虑在广义上承认椭圆形PDE的域：我们算作满足容量密度条件的1侧NTA（又名统一）域的示例，1侧和弦域域，具有低维度的ahlfors-ahlfors-david david界限，某些域的域，以及具有混合范围的某些范围的域；因此，我们的方法为椭圆运算符的Carleson扰动理论提供了统一的观点。 我们的证明不会引入锯齿状域或外推法。我们还向一些Dahlberg-kenig-Pipher操作员，自由边界问题提供了多个应用程序，我们在椭圆度措施中提供了$ a _ {\ infty} $的新特征。
We extend in two directions the notion of perturbations of Carleson type for the Dirichlet problem associated to an elliptic real second-order divergence-form (possibly degenerate, not necessarily symmetric) elliptic operator. First, in addition to the classical perturbations of Carleson type, that we call additive Carleson perturbations, we introduce scalar-multiplicative and antisymmetric Carleson perturbations, which both allow non-trivial differences at the boundary. Second, we consider domains which admit an elliptic PDE in a broad sense: we count as examples the 1-sided NTA (a.k.a. uniform) domains satisfying the capacity density condition, the 1-sided chord-arc domains, the domains with low-dimensional Ahlfors-David regular boundaries, and certain domains with mixed-dimensional boundaries; thus our methods provide a unified perspective on the Carleson perturbation theory of elliptic operators. Our proofs do not introduce sawtooth domains or the extrapolation method. We also present several applications to some Dahlberg-Kenig-Pipher operators, free-boundary problems, and we provide a new characterization of $A_{\infty}$ among elliptic measures.
